What's special about property taxes that they should be held constant? A person could just as easily be pushed out of their house by a general rise in the cost of living in their area, or increasing traffic, or a regional economic collapse. Shall we legislate those to be constant as well?
If the local government is well run, then an individual's property taxes should approximate the cost of providing local government services to them, regardless of the value of their house. If real estate values in a local jurisdiction double without changing the government's costs, property tax rates should be cut in half to compensate. I realize that there's a big "if" when it comes to the "well run" part, but if that's not being done, then you need to fight over the "well run" part, not giving people massively different tax rates based purely on when they bought their house.
I don't think this idea is all that outlandish either. My understanding is that a lot of jurisdictions work this way, either explicitly or effectively through the legislative process. A budget is set, required revenue decided upon, and property tax rates set such that the required amount is obtained. If values go up, rates fall, and vice versa.
This doesn't solve the problem of hyper-local changes, like if just one section of a county becomes extremely popular and prices are driven through the roof, while the rest is unaffected. But again, what's the difference between being driven out due to that and having, say, the local food stores double their prices to to huge demand and drive you out that way?